Integrals with Logarithms
2026ReferenceSergey
Reference table of indefinite integrals of expressions containing the natural logarithm.
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230
Search results for “natural logarithm”
22 results
A non-Noetherian local ring with finitely generated maximal ideal
2026NotesBogdan Zavyalov
A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.
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340
Fun example: empty colimit does not commute with empty limit
2026NotesBogdan Zavyalov
Why the definition of a filtered category insists on non-emptiness: the empty colimit fails to commute with the empty limit, and that is the only failure.
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310
Determinant via Permutations
2026TheorySergey
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
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170
An example of a non-reduced Picard scheme
2026TheorySean Cotner
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
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170
Finite flat commutative group schemes embed locally into abelian schemes
2026TheorySean Cotner
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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150
The torsion component of the Picard scheme
2026OtherBogdan Zavyalov
Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.
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500